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In this article, we study the recent development of the qualitative uncertainty principle on certain Lie groups. In particular, we consider that if the Weyl transform on certain step-two nilpotent Lie groups is of finite rank, then the function has to be zero almost everywhere as long as the nonvanishing set for the function has finite measure. Further, we consider that if the Weyl transform of each Fourier–Wigner piece of a suitable function on the Heisenberg motion group is of finite rank, then the function has to be zero almost everywhere whenever the nonvanishing set for each Fourier–Wigner piece has finite measure. [ABSTRACT FROM AUTHOR]

In this article, the nonclassical transient heat propagation process in a cracked strip is investigated by Guyer–Krumhansl (G–K) model, which incorporates both the time lagging behavior and the spatially nonlocal effect. The impulsive thermal loading as well as cyclic loading exerted on the top bounding surface are examined to explore the non-Fourier thermal characteristics. By means of the Laplace transform and Fourier transform, the governing partial differential equations subjected to mixed boundary conditions are converted to a group of singular integral equations. With the aid of numerical Laplace inversion, the transient temperatures are calculated to make comparisons of thermal responses determined by Fourier's law, Cattaneo–Vernotte (C–V) equation, and G–K model. The numerical results display the specific thermal behaviors of G–K model in the cracked medium and demonstrate the G–K model's capabilities in eliminating the unrealistic phenomena accompanied by C–V equation. Our research would contribute to achieving a better understanding of the transient heat conduction in small-sized systems or composites at the macroscopic scale. [ABSTRACT FROM AUTHOR]

We view the (real) Laplace transform through the lens of linear algebra as a continuous analogue of the power series by a negative exponential transformation that switches the basis of power functions to the basis of exponential functions. This approach immediately points to how the complex Laplace transform is a generalisation of the Fourier transform where the pole of the transform realises the linear algebraic intuition. The exponential transformation also motivates the Taylor inversion of the real Laplace transform. [ABSTRACT FROM AUTHOR]

Using the data from weather observation stations, an assessment of statistical significance of the first harmonic amplitude of the long-term average daily precipitation series in Russia in 1961‒2020 and their changes in 1991‒2020 compared to 1961‒1990 is made. It is shown that at most stations the first harmonic has a reliable value different from the noise, except for eleven stations in the southern regions of the European part of Russia. It is revealed that on average there is a significant decrease in the first harmonic amplitudes of the daily precipitation series across Russia. An analysis of the spatial distribution of the studied quantities is performed and the presence of large areas with their homogeneous character is demonstrated. [ABSTRACT FROM AUTHOR]

As the number of electronic gadgets in our daily lives is increasing and most of them require some kind of human interaction, this demands innovative, convenient input methods. There are limitations to state-of-the-art (SotA) ultrasound-based hand gesture recognition (HGR) systems in terms of robustness and accuracy. This research presents a novel machine learning (ML)-based end-to-end solution for hand gesture recognition with low-cost micro-electromechanical (MEMS) system ultrasonic transducers. In contrast to prior methods, our ML model processes the raw echo samples directly instead of using pre-processed data. Consequently, the processing flow presented in this work leaves it to the ML model to extract the important information from the echo data. The success of this approach is demonstrated as follows. Four MEMS ultrasonic transducers are placed in three different geometrical arrangements. For each arrangement, different types of ML models are optimized and benchmarked on datasets acquired with the presented custom hardware (HW): convolutional neural networks (CNNs), gated recurrent units (GRUs), long short-term memory (LSTM), vision transformer (ViT), and cross-attention multi-scale vision transformer (CrossViT). The three last-mentioned ML models reached more than 88% accuracy. The most important innovation described in this research paper is that we were able to demonstrate that little pre-processing is necessary to obtain high accuracy in ultrasonic HGR for several arrangements of cost-effective and low-power MEMS ultrasonic transducer arrays. Even the computationally intensive Fourier transform can be omitted. The presented approach is further compared to HGR systems using other sensor types such as vision, WiFi, radar, and state-of-the-art ultrasound-based HGR systems. Direct processing of the sensor signals by a compact model makes ultrasonic hand gesture recognition a true low-cost and power-efficient input method. [ABSTRACT FROM AUTHOR]

Ocean Acoustic Waveguide Remote Sensing (OAWRS) typically utilizes large-aperture linear arrays combined with coherent beamforming to estimate the spatial distribution of acoustic scattering echoes. The conventional maximum likelihood deconvolution (DCV) method uses a likelihood model that is inaccurate in the presence of multiple adjacent targets with significant intensity differences. In this study, we propose a deconvolution algorithm based on a modified likelihood model of beamformed intensities (M-DCV) for estimation of the spatial intensity distribution. The simulated annealing iterative scheme is used to obtain the maximum likelihood estimation. An approximate expression based on the generalized negative binomial (GNB) distribution is introduced to calculate the conditional probability distribution of the beamformed intensity. The deconvolution algorithm is further simplified with an approximate likelihood model (AM-DCV) that can reduce the computational complexity for each iteration. We employ a direct deconvolution method based on the Fourier transform to enhance the initial solution, thereby reducing the number of iterations required for convergence. The M-DCV and AM-DCV algorithms are validated using synthetic and experimental data, demonstrating a maximum improvement of 73% in angular resolution and a sidelobe suppression of 15 dB. Experimental examples demonstrate that the imaging performance of the deconvolution algorithm based on a linear small-aperture array consisting of 16 array elements is comparable to that obtained through conventional beamforming using a linear large-aperture array consisting of 96 array elements. The proposed algorithm is applicable for Ocean Acoustic Waveguide Remote Sensing (OAWRS) and other sensing applications using linear arrays. [ABSTRACT FROM AUTHOR]

The main aim of this article is to derive certain continuity and boundedness properties of the coupled fractional Fourier transform on Schwartz-like spaces. We extend the domain of the coupled fractional Fourier transform to the space of tempered distributions and then study the mapping properties of pseudo-differential operators associated with the coupled fractional Fourier transform on a Schwartz-like space. We conclude the article by applying some of the results to obtain an analytical solution of a generalized heat equation. [ABSTRACT FROM AUTHOR]

The thermoreflectance method, which can measure thermal diffusivity in the cross-plane direction of thin films, mainly has two possible configurations; rear-heat front-detect (RF) and front-heat front-detect (FF) configuration. FF configuration is applicable to a wide variety of thin films including thin films deposited on opaque substrates, but this configuration has some problems in determination of the thermal diffusivity. One of the main problems is the effect of the penetration of pump beam and probe beam in thin film, which affects the initial temperature distribution near the sample's surface after pulse heating. Several studies have tried to analyze the effect but there have been no practical analytical solutions which can solve this problem in FF configuration. In this paper, we propose a new analytical solution which considers the penetration of pump beam and probe beam into thin film, and by applying Fourier expansion analysis which we developed in a previous study to thermoreflectance signals, we have determined the thermal diffusivity of thin film in the thermoreflectance method under FF configuration. We measured platinum thin films with different thickness under both FF and RF configuration and obtained consistent thermal diffusivity values from both configurations. [ABSTRACT FROM AUTHOR]

In this paper, we derive some Fourier transforms of confluent hypergeometric functions. We give generalizations of several well-known results involving Fourier transforms of gamma functions. In particular, the generalizations include some Ramanujan's remarkable formulas. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of phase retrieval with short-time linear canonical transform (STLCT). The relation between signal, window and their STLCT is provided through Fourier transform. Based on this theorem, a uniqueness result is established for all square integrable functions. For nonseparable real continuous signal, we prove the uniqueness theorems under some weaker conditions. In complex bandlimited and cardinal B-spline spaces, uniqueness results are provided with magnitude-only STLCT. [ABSTRACT FROM AUTHOR]

In order to realize the real-time processing and analysis of astronomical ultrawide bandwidth signals, this study proposes a sub-band division algorithm based on RFSoC. The algorithm uses Kaiser window to design FIR prototype low-pass filter, adopts critical sampling polyphase filter bank to decompose ultra-wide bandwidth signal into several sub-bands, and encapsulates each sub-band into VDIF data frame and sends it to GPU server. The algorithm is implemented on RFSoC platform, and its effectiveness is verified by simulation and actual observation. The experimental results show that the algorithm can divide the astronomical ultra-wide bandwidth signal into multiple sub-bands in real time, packetize and transmit them to GPU. This research provides reproducible design and project for ultra-wide bandwidth signal sub-band division with low spectrum leakage and aliasing, high data accuracy, and fast processing speed. [ABSTRACT FROM AUTHOR]

The classical Eshelby's theory, developed based on local linear elasticity, cannot be applied to inclusion problems that involve nonlocal (long range) elastic effects often observed in micromechanical systems. In this study, we introduce the extension of Eshelby's inclusion theory to nonlocal elasticity. Starting from Eringen's integral formulation of nonlocal elasticity, an eigenstress-driven nonlocal Eshelby's inclusion theory is presented. The eigenstress-driven approach is shown to be a valid mathematical extension of the classical eigenstrain-driven approach in the context of nonlocal inclusion problems. Two individual numerical approaches are developed and applied to simulate inclusion problems and numerically extract the corresponding nonlocal Eshelby tensor. The numerical results obtained from both approaches confirm the validity of the derived nonlocal Eshelby tensor and its ability to capture the non-uniform eigenstress distribution within an elliptic inclusion. These results also help reveal the fundamental difference between the mechanical behaviour of the classical local and the nonlocal inclusion problems. The eigenstress-driven nonlocal inclusion theory could provide the necessary theoretical foundation for the development of homogenization methods of nonlocal heterogeneous media. [ABSTRACT FROM AUTHOR]

AbstractHigh-throughput mechanical testing based on functionally graded specimens is very promising for accelerating the development of new materials. However, due to the inhomogeneity-induced complexity, most existing analyses on functionally graded materials have recourse to numerical methods to predict their mechanical responses in reaction to external stimuli. This work investigates the surface Green’s functions for an inhomogeneous half-plane with horizontal exponential material gradient subject to both normal and tangential concentrated forces acting on the surface. The governing equations are first simplified by introducing appropriate potential functions, which facilitates the mathematical derivation of displacements via the Fourier transform technique. In the case of normal force, the vertical surface displacement is derived explicitly under the weak gradient assumption while the horizontal surface displacement is derived directly without the same assumption. In particular, the Meijer G-function and Fox H-function are introduced to express and simplify the vertical displacement. In the case of tangential force, the analytical expressions of surface displacements are also derived similarly. It is noted that the surface Green’s functions not only exhibit singularity and asymmetry properties as expected, but also can be reduced to the classical Boussinesq-Flamant solutions for a homogeneous half-plane. In addition, the analytical results are verified through comparison with the finite element analyses. The surface Green’s functions derived here could be a theoretical basis for developing high-throughput mechanical testing methods which use specimens made of functionally graded materials. [ABSTRACT FROM AUTHOR]

In this present work, we establish new fundamental uncertainty principles for the fractional ambiguity function. These principles are generalized forms of inequalities concerning the Fourier transform. [ABSTRACT FROM AUTHOR]

In this paper, we consider a problem of estimating a large loss probability of financial derivatives portfolio, which are commonly modeled as nested expectations. However, the cost of nested simulation may be too expensive and thus multilevel Monte Carlo (MLMC) method is recently used to reduce the nested simulation complexity. When using antithetic MLMC to solve the indicator function, we get the complexity of O(ϵ -5/2). To decrease the computational burden, we use a Fourier transform method to modify the form of indicator function. The new estimator is sufficiently smooth and enables the antithetic MLMC method to achieve a better complexity. In addtion, we combine quasi-Monte Carlo (QMC) with MLMC to reduce the variance of inner estimator. Numerical results show that using the Fourier transform method in both MLMC and MLQMC can attain the optimal complexity O(ϵ -2). [ABSTRACT FROM AUTHOR]

Research on knocking combustion is vitally essential since it impacts engine performance regarding noise and emissions, fuel consumption, power density, and durability. Because of end-gas auto-ignition, current SI engines may experience conventional knock, which restricts increasing the compression ratio to increase thermal efficiency. Therefore, this paper examined the in-cylinder pressure and vibration of a port fuel-injected gasoline engine under knocking and non-knocking conditions. The heat release rate was calculated from pressure traces where the crank angle resolutions were not constant, and the conversion from the time domain to the frequency domain of knock sensor signals by using the Fourier Transform was performed. The result reveals that more heat is released during knocking combustion than during non-knocking combustion. When there is a knock, the temperature variation increases. This would facilitate the use of heat transfer studies to detect knock occurrences. Moreover, knocking is caused by an ignition timing that is too advanced, 22°BTC, compared to the non-knocking spark case, 14°BTC. In the frequency domain, the amplitude of the knocking combustion is higher than the amplitude of the non-knocking combustion, indicating the engine’s noise and vibration. [ABSTRACT FROM AUTHOR]

Toward the goal of minimizing construction costs while maintaining high performance, a new, reduced-pressure, drift tube ion mobility system is coupled with an ion trap mass analyzer through a custom ion shuttle. The availability of reduced-pressure ion mobility systems remains limited due to comparatively expensive commercial options and limited shared design features in the open literature. This report details the complete design and benchmarking characteristics of a reduced-pressure ion mobility system. The system is constructed from FR4 PCB electrodes and encased in a PTFE vacuum enclosure with custom torque-tightened couplers to utilize standard KF40 bulkheads. The PTFE enclosure directly minimizes the overall system expenses, and the implementation of threaded brass inserts allows for facile attachments to the vacuum enclosure without damaging the thermoplastic housing. Front and rear ion funnels maximize ion transmission and help mitigate the effects of radial ion diffusion. A custom planar ion shuttle transports ions from the exit of the rear ion funnel into the ion optics of an ion trap mass analyzer. The planar ion shuttle can couple the IM system to any contemporary Thermo Scientific ion trap mass analyzer. Signal stability and ion intensity remain unchanging following the implementation of the planar ion shuttle when compared to the original stacked ring ion guide. The constructed IM system showed resolving powers up to 85 for various small molecules and proteins using the Fourier transform from a ∼1 m drift tube. Recorded mobilities derived from first principles agree with published literature results with an average error of 1.1% and an average error toward literature values using single field calibration of <1.3%. [ABSTRACT FROM AUTHOR]

In an effort to estimate the number of true nulls in large scale multiplicity problems (the normal means problem), we generalize the current Fourier transform based oracle estimator with a Laplace transform based estimator. Our interest in this problem stems from the application of r-power which requires knowledge of the number of nulls (Dasgupta et al. in Sankhya B 78(1):96–118, 2016). We analytically show that our method is consistent and theoretically has lower mean squared error than the existing competitor (Jin in J R Stat Soc Ser B (Stat Methodol) 70(3):461–493, 2008). We follow up by a numerical example and a simulation study that ratifies our theoretical results. [ABSTRACT FROM AUTHOR]

For the octonion Fourier transform, we establish the real Paley‐Wiener theorem. It relates the mean of derivatives of a function with the support of its octonion Fourier transform via limm→∞‖∂mαf‖L2(ℝ3,핆)1/m=(2π)α‖wα‖L∞(suppF핆{f},핆)for any octonion‐valued Sobolev function f∈H∞(ℝ3,핆). [ABSTRACT FROM AUTHOR]

Background: Alterations in sleep have been described in multiple health conditions and as a function of several medication effects. However, evidence generally stems from small univariate studies. Here, we apply a large-sample, data-driven approach to investigate patterns between in sleep macrostructure, quantitative sleep EEG, and health. Methods: We use data from the MrOS Sleep Study, containing polysomnography and health data from a large sample (N = 3086) of elderly American men to establish associations between sleep macrostructure, the spectral composition of the electroencephalogram, 38 medical disorders, 2 health behaviors, and the use of 48 medications. Results: Of sleep macrostructure variables, increased REM latency and reduced REM duration were the most common findings across health indicators, along with increased sleep latency and reduced sleep efficiency. We found that the majority of health indicators were not associated with objective EEG power spectral density (PSD) alterations. Associations with the rest were highly stereotypical, with two principal components accounting for 85–95% of the PSD-health association. PC1 consists of a decrease of slow and an increase of fast PSD components, mainly in NREM. This pattern was most strongly associated with depression/SSRI medication use and age-related disorders. PC2 consists of changes in mid-frequency activity. Increased mid-frequency activity was associated with benzodiazepine use, while decreases were associated with cardiovascular problems and associated medications, in line with a recently proposed hypothesis of immune-mediated circadian demodulation in these disorders. Specific increases in sleep spindle frequency activity were associated with taking benzodiazepines and zolpidem. Sensitivity analyses supported the presence of both disorder and medication effects. Conclusions: Sleep alterations are present in various health conditions. [ABSTRACT FROM AUTHOR]

The aim of this work is to derive a symbol calculus on L 2 (R n) for multidimensional Hausdorff operators. Two aspects of this activity result in two almost independent parts. While throughout the perturbation matrices are supposed to be self-adjoint and form a commuting family, in the second part they are additionally assumed to be positive definite. What relates these two parts is the powerful method of diagonalization of a normal Hausdorff operator elaborated earlier by the second named author. [ABSTRACT FROM AUTHOR]

Given a dilation matrix M, a so-called space of M-positive vectors in the Euclidean space is introduced and studied. An algebraic structure of this space is similar to the positive half-line equipped with the termwise addition modulo 2, which is used in the Walsh analysis. The role of harmonics is played by some analogues of the classical Walsh functions. The concept of Fourier transform is introduced, and the Poisson summation formula, Plancherel theorem, Vilenkin-Chrestenson formulas and so on are proved. A kind of analogue of the Schwartz class is studied. This class consists of functions such that both the function itself and its Fourier transform have compact support. [ABSTRACT FROM AUTHOR]

The Dirichlet problem for a multidimensional elliptic system in the half-space is considered. With the help of the Fourier transform, the problem of the solvability of the problem is reduced to the study of a second-order partial differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the structural properties of Nevanlinna measures, that is, Borel measures that arise in the integral representation of Herglotz–Nevanlinna functions. In particular, we give a characterization of these measures in terms of their Fourier transform, characterize measures supported on hyperplanes including extremal measures, describe the structure of the singular part of the measures when some variable are set to a fixed value, and provide estimates for the measure of expanding and shrinking cubes. Corresponding results are stated also in the setting of the polydisc where applicable, and some of our proofs are actually performed via the polydisc. [ABSTRACT FROM AUTHOR]

Gas turbines play a vital role in gas transportation and power generation, but they are prone to instability phenomena that can lead to vibrations, shorten equipment lifespan, and result in catastrophic failures. To tackle these challenges, a paper introduces an integrated approach that leverages advanced techniques like Fourier transform, Neuro-Fuzzy systems, and wavelet analysis for continuous monitoring of the MS5002C turbine's condition. The proposed method begins by collecting operational data and utilizing the Fourier transform to measure vibratory quantities, accurately representing their evolution through spectral data obtained from the analyzed signals. Adaptive inference-based algorithms of neuro-fuzzy systems are then employed to generate turbine failure indicators. This approach enables the development of a model-based fault detection method that compares the actual turbine operation with the estimated operation derived from a pre-established model, enabling the classification of detected faults. To enhance decision-making quality, evaluation, and validation of the diagnostic strategy's performance, a multi-resolution analysis based on the wavelet transform is applied. The presented results from various implementation and validation tests demonstrate the effectiveness of this intelligent diagnostic approach in detecting and analyzing gas turbine vibrations. The paper exhibits promising outcomes in real-time monitoring, ensuring the operational safety of the turbine. [ABSTRACT FROM AUTHOR]

Let { A n } n = 1 ∞ be a sequence of expanding matrices with A n ∈ M 2 ⁢ (ℤ) , and let { D n } n = 1 ∞ be a sequence of three-element digit sets with { x ∈ (0 , 1) 2 : ∑ d ∈ D n e 2 ⁢ π ⁢ i ⁢ 〈 d , x 〉 = 0 } = { ± 1 3 ⁢ (1 , i) t } , i ∈ { 1 , 2 } . The associated Moran measure generated by the infinite convolution μ { A n } , { D n } = δ A 1 - 1 ⁢ D 1 * δ A 1 - 1 ⁢ A 2 - 1 ⁢ D 2 * δ A 1 - 1 ⁢ A 2 - 1 ⁢ A 3 - 1 ⁢ D 3 * ⋯. In this paper, we give some necessary and sufficient conditions for μ { A n } , { D n } to be a spectral measure under some suitable conditions on A n and D n . [ABSTRACT FROM AUTHOR]

Let 0 < ρ < 1 and let { a j , b j , n j } j = 1 ∞ be a sequence of positive integers with an upper bound. Associated with them, there exists a unique Borel probability measure μ ρ , { 0 , a j , b j } , { n j } generated by the following infinite convolution of discrete measures: μ ρ , { 0 , a j , b j } , { n j } = δ ρ n 1 ⁢ { 0 , a 1 , b 1 } ∗ δ ρ n 1 + n 2 ⁢ { 0 , a 2 , b 2 } ∗ δ ρ n 1 + n 2 + n 3 ⁢ { 0 , a 3 , b 3 } ∗ ⋯ , where gcd ⁡ (a j , b j) = 1 for all j ∈ ℕ . In this paper, we show that L 2 ⁢ (μ ρ , { 0 , a j , b j } , { n j } ) admits an exponential orthonormal basis if and only if the following two conditions are satisfied: (i) { a j , b j } ≡ { ± 1 } ⁢ (mod ⁢ 3) for all j ≥ 1 ; (ii) there exists a natural number r such that ρ - r ∈ 3 ⁢ ℕ and n j ∈ r ⁢ ℕ for all j ≥ 2 . [ABSTRACT FROM AUTHOR]

To further enhance the security of image encryption, a new chaos-based image encryption algorithm (IEA) based on discrete Fourier transform and Joseph traversal is proposed to encrypt the plain image in both the frequency domain and space domain simultaneously. In the proposed IEA, the logistic map is used to generate the appropriate chaotic sequence, and the improved Joseph traversal is used to scramble the image in different starting positions and variable step sizes. Then, block diffusion is performed at the end. The main finding concerning the proposed IEA is that the combination of discrete Fourier transform and Joseph traversal can enhance the security of the image information, which has been validated by measuring the performance in resisting the common types of attacks. [ABSTRACT FROM AUTHOR]

Background: Currently, determining procedural endpoints and treatment efficacy of vascular interventions is largely qualitative and relies on subjective visual assessment of digital subtraction angiography (DSA) images leading to large interobserver variabilities and poor reproducibility. Quantitative metrics such as the residual blood velocity in embolized vessel branches could help establish objective and reproducible endpoints. Recently, velocity quantification techniques based on a contrast enhanced X‐ray sequence such as qDSA and 4D DSA have been proposed. These techniques must be robust, and, to avoid radiation dose concerns, they should be compatible with low dose per frame image acquisition. Purpose: To develop and evaluate a technique for robust blood velocity quantification from low dose contrast enhanced X‐ray image sequences that leverages the oscillating signal created by pulsatile blood flow. Methods: The proposed spatiotemporal frequency domain (STF) approach quantifies velocities from time attenuation maps (TAMs) representing the oscillating signal over time for all points along a vessel centerline. Due to the time it takes a contrast bolus to travel along the vessel centerline, the resulting TAM resembles a sheared sine wave. The shear angle is related to the velocity and can be determined in the spatiotemporal frequency domain after applying the 2D Fourier transform to the TAM. The approach was evaluated in a straight tube phantom using three different radiation dose levels and compared to ultrasound transit‐time‐based measurements. The STF velocity results were also compared to previously published approaches for the measurement of blood velocity from contrast enhanced X‐ray sequences including shifted least squared (SLS) and phase shift (PHS). Additionally, an in vivo porcine study (n = 8) was performed where increasing amounts of embolic particles were injected into a hepatic or splenic artery with intermittent velocity measurements after each injection to monitor the resulting reduction in velocity. Results: At the lowest evaluated dose level (average air kerma rate 1.3 mGy/s at the interventional reference point), the Pearson correlation between ultrasound and STF velocity measurements was 99%$99\%$. This was significantly higher (p<0.0001$p < 0.0001$) than corresponding correlation results between ultrasound and the previously published SLS and PHS approaches (91$\hskip.001pt 91$ and 93%$93\%$, respectively). In the in vivo study, a reduction in velocity was observed in 85.7%$85.7\%$ of cases after injection of 1 mL, 96.4%$96.4\%$ after 3 mL, and 100.0%$100.0\%$ after 4 mL of embolic particles. Conclusions: The results show good agreement of the spatiotemporal frequency domain approach with ultrasound even in low dose per frame image sequences. Additionally, the in vivo study demonstrates the ability to monitor the physiological changes due to embolization. This could provide quantitative metrics during vascular procedures to establish objective and reproducible endpoints. [ABSTRACT FROM AUTHOR]

In the realm of neural network-based noise reduction, conventional models predominantly address Gaussian and blur artifacts across entire images. However, they encounter notable challenges when directly applied to periodic noise characteristics of high-resolution infrared sequential imagery. The high resolution also complicates the construction of suitable datasets. Our study introduces an innovative strategy that transforms two-dimensional images into one-dimensional signals, eliminating the need for processing the full image. We have developed a simulated dataset that closely mirrors natural infrared line scanning images derived from the FLIR dataset. To address low-frequency periodic noise, we propose two neural-network-based denoising approaches. The first employs a neural network to deduce noise from the one-dimensional signal, while the second utilizes discrete Fourier transforms for noise prediction within the frequency domain. Our experimental results highlight the Restormer model's exemplary performance in direct noise prediction, where denoised images attain a PSNR of around 41 and an SSIM close to 0.9 on simulated data, leaving minimal residual noise in the actual denoised images. Further, we investigate the influence of Fourier coefficients, as predicted by neural networks, on the denoising process in the second approach. Employing 12 frequency coefficients, the Restormer and NAFNet models both reach a PSNR near 34 and an SSIM of approximately 0.842. [ABSTRACT FROM AUTHOR]

In this work, we propose a morphologic analysis by means of the construction of 3D models of the SARS-CoV-2 VP (viral particles) with algorithms in Python and Matlab based on the processing of frames. To this aim, we simulate the Takeda method to induce periodicity and apply the Fourier transform to obtain the phase of objects under analysis. To this aim, we analyze several research works focused on infected tissues by SARS-CoV-2 virus culture cells, highlighting the obtained medical images of the virus from microscopy and tomography. We optimize the results by performing image processing (segmentation and periodic noise removal) in order to obtain an accurate ROI (Region of Interest) segmentation containing only information on SARS-CoV-2 cells. We apply our algorithm to these images (3D tomographic medical images) to simulate the Takeda method (which also filters the image), considering the periodicity induced by us in the image to carry out a phase unwrapping process. Finally, we use the image phase to focus on the body, center (RNA, Protein M-N), and spikes (Protein S) of the SARS-CoV-2 cells to identify them as characteristic elements of the SARS-CoV-2 virion morphology to build a 3D model based only in the metadata of clinical studies on cell cultures. The latter results in the construction of a mathematical, physical, biological, and numerical model of the SARS-CoV-2 virion, a tool with volumes, or 3D non-speculative or animated models, based only on medical images (3D tomography) in clinical tests, faithful to the virus. [ABSTRACT FROM AUTHOR]

Given a one-parameter family of continuous linear operators , with , we consider the optimal recovery of the values of on the whole space by approximate information on the values of , where runs over a compact set and . We find a family of optimal methods for recovering the values of . Each of these methods uses approximate measurements at no more than two points in and depends linearly on these measurements. As a corollary, we provide some families of optimal methods for recovering the solution of the heat equation at a given moment of time from inaccurate measurements on other time intervals and for solving the Dirichlet problem for a half-space on a hyperplane by inaccurate measurements on other hyperplanes. The optimal recovery of the values of from the indicated information reduces to finding the value of an extremal problem for the maximum with continuum many inequality-type constraints, i.e., to finding the exact upper bound of the maximized functional under these constraints. This rather complicated task reduces to the infinite-dimensional problem of linear programming on the vector space of all finite real measures on the -algebra of Lebesgue measurable sets in . This problem can be solved by some generalization of the Karush–Kuhn–Tucker theorem, and its significance coincides with the significance of the original problem. [ABSTRACT FROM AUTHOR]

We find explicit expressions for optimal recovery methods in the problem of recovering the values of continuous linear operators on a Sobolev function class from the following information: The Fourier transform of functions is known approximately on some measurable subset of the finite-dimensional space on which the functions are defined. As corollaries, we obtain optimal methods for recovering the solution to the heat equation and solving the Dirichlet problem for a half-space. [ABSTRACT FROM AUTHOR]

In this note we study character sheaves for graded Lie algebras arising from inner automorphisms of special linear groups and Vinberg's type II classical graded Lie algebras. [ABSTRACT FROM AUTHOR]

Real-world data often exhibits a long-tailed distribution in practical scenarios. However, deep learning models usually face challenges when it comes to effectively identifying infrequent classes amidst the abundance of prevalent ones. The fundamental issue lies in the scarcity of available information for tail classes. A highly intuitive approach is to uncover a greater amount of valuable information specifically tailored to these tail classes. We find that textual information of class names and frequency domain information of images are ignored by previous works in long-tailed visual recognition. Therefore, we propose a Text-Guided Fourier Augmentation (TGFA) method with the aid of language models and the Fourier transform to excavate more useful information for tail classes. Extensive experiments demonstrate that our proposed method effectively enriches training data on-the-fly, allowing for an end-to-end one-stage supervised contrastive learning framework that surpasses other methods including two-stage or multi-experts methods, in terms of efficiency and performance. • A novel Text-Guided Fourier Augmentation method for long-tailed visual recognition. • First work fusing textual and Fourier information to enhancing long-tailed learning. • Our end-to-end method outperforms two-stage or multi-experts long-tailed methods. • Superior over other methods in terms of efficiency and performance. [ABSTRACT FROM AUTHOR]

This paper proposes a frequency-time hybrid solver for the time-dependent wave equation in two-dimensional interior spatial domains. The approach relies on four main elements, namely, (1) A multiple scattering strategy that decomposes a given interior time-domain problem into a sequence of limited-duration time-domain problems of scattering by overlapping open arcs, each one of which is reduced (by means of the Fourier transform) to a sequence of Helmholtz frequency-domain problems ; (2) Boundary integral equations on overlapping boundary patches for the solution of the frequency-domain problems in point (1); (3) A smooth "Time-windowing and recentering" methodology that enables both treatment of incident signals of long duration and long time simulation; and, (4) A Fourier transform algorithm that delivers numerically dispersionless, spectrally-accurate time evolution for given incident fields. By recasting the interior time-domain problem in terms of a sequence of open-arc multiple scattering events, the proposed approach regularizes the full interior frequency domain problem—which, if obtained by either Fourier or Laplace transformation of the corresponding interior time-domain problem, must encapsulate infinitely many scattering events, giving rise to non-uniqueness and eigenfunctions in the Fourier case, and ill conditioning in the Laplace case. Numerical examples are included which demonstrate the accuracy and efficiency of the proposed methodology. [ABSTRACT FROM AUTHOR]

For the fourth-order partial differential equation proposed by Fushchich for mathematical description of heat and mass transfer processes, the class of uniqueness and the class of correctness of the Cauchy problem have been established. Using the Fourier transform, a solution to the Cauchy problem is found in an explicit analytical form. It is shown that the new equation does not improve the properties of the classical heat conduction equation, but it allows one to describe heat transfer processes taking into account relaxation effects within the framework of the Galileo-invariant mathematical theory of transfer. [ABSTRACT FROM AUTHOR]

In this paper, we define generalized Fourier and inverse Fourier transforms containing the h-exponential function in their kernels and give the fundamental properties of these transforms. We also compute the transforms of both the classical and the generalized Riemann-Liouville and Caputo fractional operators. In addition, we compute the transforms of some elementary and generalized special functions as well. Finally, as applications, we obtain the solutions of two differential equations with ordinary and fractional derivatives using the transforms we have defined. [ABSTRACT FROM AUTHOR]

New classes of evolution differential equations with the Liouville derivative in Banach spaces are studied. Equations are considered on the whole real line and are not endowed by the initial conditions. Using the methods of the Fourier transform theory, we prove the unique solvability in the sense of classical solutions for the equation solved with respect to the Liouville fractional derivative with a bounded operator at the unknown function. This allows us to obtain the analogous result for the equation with a linear degenerate operator at the fractional derivative and with a spectrally bounded pair of operators. Abstract results are applied to obtain new results on the unique solvability of systems of ordinary differential equations, boundary problems to partial differential equations, and systems of equations. [ABSTRACT FROM AUTHOR]

From the beginning of COVID-19 pandemic, numerous mutants of SARS-CoV-2 have since been evolved owing to high transmissibility and virulence. Due to the limited effectiveness of previously imposed vaccines and preventive therapies, these strains are still causing concern. This paper proposes comparative evaluation of three novel genomic signal processing-based methods employing discrete wavelet decomposition with lifting (DWT), discrete Fourier transform (DFT), and singular value decomposition (SVD) for the classification of emerging SARS-CoV-2 variants utilizing feature extraction from collected SARS-CoV-2 variants acquired from the NCBI virus database. The efficiency and accuracy of the proposed alignment-free algorithms have been tested using three Coronavirus datasets including human Coronavirus (HCoV), SARS-CoV-2 variants (CoV-Variants and Omicron). The viral nucleotide sequences which are converted into numerical representation leveraging purine-pyrimidine mapping, DNA walk & Z-curve are fed into DWT, SVD, & DFT processors, respectively. In the approach with DWT, the second-generation wavelet transform employs two best wavelet bases Daubechies (Db) and Biorthogonal (Bior) based on the validation of the HCoV dataset for the feature extraction of the CoV-Variants dataset. Various machine learning algorithms, such as Support Vector Machine, K-nearest neighbors, and ensemble, are used to classify the virus strains and evaluate the efficacy of the algorithm. Finally, hyper-parametric tuning is done utilizing the Bayesian optimization technique to select the best fit model for KNN and SVM. The proposed algorithm has successfully classified the CoV-Variants dataset with an average accuracy of 98.76% utilizing the DWT, DFT, and SVD, while the best-achieved accuracy for this dataset is 98.9% using the DWT technique employing purine–pyrimidine mapping. The best-achieved accuracy rate for predicting Omicron is 99.8% using SVD-based technique. The best-obtained accuracy for HCoV dataset is 100% resulted in all three methods. [ABSTRACT FROM AUTHOR]

This paper introduces a method for high-resolution lattice image reconstruction and dislocation analysis based on diffraction extinction. The approach primarily involves locating extinction spots in the Fourier transform spectrum (reciprocal space) and constructing corresponding diffraction wave functions. By the coherent combination of diffraction and transmission waves, the lattice image of the extinction planes is reconstructed. This lattice image is then used for dislocation localization, enabling the observation and analysis of crystal planes that exhibit electron diffraction extinction effects and atomic jump arrangements during high-resolution transmission electron microscopy (HRTEM) characterization. Furthermore, due to the method's effectiveness in localizing dislocations, it offers a unique advantage when analyzing high-resolution images with relatively poor quality. The feasibility of this method is theoretically demonstrated in this paper. Additionally, the method was successfully applied to observed edge dislocations, such as 1 / 6 [ 21 1 − ] , 1 / 6 [ 2 − 1 1 − ] , and 1 / 2 [ 0 1 − 1 ] , which are not easily observable in conventional HRTEM characterization processes, in electro-deposited Cu thin films. The Burgers vectors were determined. Moreover, this paper also attempted to observe screw dislocations that are challenging to observe in high-resolution transmission electron microscopy. By shifting a pair of diffraction extinction spots and superimposing the reconstructed images before and after the shift, screw dislocations with a Burgers vector of 1 / 2 [ 01 1 − ] were successfully observed in electro-deposited Cu thin films. [ABSTRACT FROM AUTHOR]

Fault diagnosis of analog circuits is a classical problem, and its difficulty lies in the similarity between fault features. To address the issue, an end-to-end mutually exclusive autoencoder (EEMEAE) fault diagnosis method for analog circuits is proposed. In order to make full use of the advantages of Fourier transform(FT) and wavelet packet transform(WPT) for extracting signal features, the original signals processed by FT and WPT are fed into two autoencoders respectively. The hidden layers of the autoencoders are mutually exclusive by Euclidean distance restriction. And the reconstruction layer is replaced by a softmax layer and 1-norm combined with cross-entropy that can effectively enhance the discriminability of features. Finally, the learning rate is adjusted adaptively by the difference of loss function to further improve the convergence speed and diagnostic performance of the model. The proposed method is verified by the simulation circuit and actual circuit and the experimental results illustrate that it is effective. [ABSTRACT FROM AUTHOR]

A major application of the inverse scattering and tomography methods is imaging all types of structural, physical, chemical and biological features of matter. The term vortex beam refers to a beam of electromagnetic radiation, electrons, photons or others—whose phase changes in corkscrew-like manner along the direction of propagation. The paper is devoted to the use of scalar Bessel beams of integer and fractional mode for the reconstruction of scattering potential. In practical applications, one naturally deals with Bessel beams truncated in the radial direction. The inversion formula for truncated Bessel beams is also obtained. Instead of the conventional Fourier diffraction theorem (Kak and Slaney in Principles of computerized tomographic imaging, SIAM, New York, 2001), the relations connecting the scattered field and the scattering potential in the Fourier space are obtained in the explicit form. [ABSTRACT FROM AUTHOR]

The thermoreflectance technique is one of the few methods which can measure thermal diffusivity of thin films as thin as 100 nm or thinner in the cross-plane direction. The thermoreflectance method under rear-heat front-detect configuration is sometimes called ultrafast laser flash method because of its similarity to laser flash method. Up to now it has typically only been possible to attempt to evaluate the interfacial thermal resistance between the thin films by preparing and measuring several samples with different thicknesses. In this study, a method to directly determine interfacial thermal resistance by a single measurement of a thin film on substrate is represented, by analyzing the shape of thermoreflectance signals with analytical solutions in frequency domain and time domain. Thermoreflectance signals observed from metallic thin films on sapphire substrate with different thickness steps were analyzed by Fourier analysis and fitted by analytical equations with four parameters: heat diffusion time across the first layer, ratio of virtual heat sources, characteristic time of cooling determined by interfacial thermal resistance and relative amplitude of the signal. Interface thermal resistance between the thin film and substrate was able to be determined reliably with smaller uncertainty. [ABSTRACT FROM AUTHOR]

Federated learning (FL) is a distributed machine learning framework that enables scattered participants to collaboratively train machine learning models without revealing information to other participants. Due to its distributed nature, FL is susceptible to being manipulated by malicious clients. These malicious clients can launch backdoor attacks by contaminating local data or tampering with local model gradients, thereby damaging the global model. However, existing backdoor attacks in distributed scenarios have several vulnerabilities. For example, (1) the triggers in distributed backdoor attacks are mostly visible and easily perceivable by humans; (2) these triggers are mostly applied in the spatial domain, inevitably corrupting the semantic information of the contaminated pixels. To address these issues, this paper introduces a frequency-domain injection-based backdoor attack in FL. Specifically, by performing a Fourier transform, the trigger and the clean image are linearly mixed in the frequency domain, injecting the low-frequency information of the trigger into the clean image while preserving its semantic information. Experiments on multiple image classification datasets demonstrate that the attack method proposed in this paper is stealthier and more effective in FL scenarios compared to existing attack methods. [ABSTRACT FROM AUTHOR]

We consider estimates for the Fourier transform of measures concentrated on smooth surfaces given by the graph of a smooth function with simple Arnold singularities such that both principal curvatures of the surface vanish at some point. We prove that if the multiplicity of the critical point of the function does not exceed , then the Fourier transforms of the corresponding surface measures belong to for any . Note that for any smooth surface the Fourier transform of a nontrivial surface measure with compact support does not belong to ; i.e., the -estimate obtained is sharp. Moreover, there exists a function with an singularity (the multiplicity of the critical point of the function is equal to ) such that the Fourier transform of the corresponding surface measure does not belong to , which shows the sharpness of the results for the multiplicity of the critical point. [ABSTRACT FROM AUTHOR]

Flexural-gravity wave blocking occurs in the presence of lateral compressive force. The existing expansion formulae for the velocity potential were limited in the case of distinct real roots of the dispersion relation with the multiplicity one. This limitation is overcome, and we obtain the complete solution for the flexuralgravity wave-maker problem at blocking and saddle points by deriving the expansion formula for velocity potential using the Fourier transform corresponding to the roots of the dispersion relation with multiplicity two and three, respectively. Further, we check the convergence of the newly derived expansion formulae by using the spectral representation of the associated eigenfunction. It has been proved that the eigenfunctions related to the expansion formulae are linearly dependent and satisfy the orthogonal modecoupling relation. Next, we solve a model problem of flexural-gravity wave scattering by a crack using the derived expansion formulae and establish the energy balance relation by applying Green's integral theorem at blocking points for infinite water depth. In addition, we have presented the validation of energy balance relation involving the generalized scattering coefficients. [ABSTRACT FROM AUTHOR]

Load forecasting is a research hotspot in academia; in the context of new power systems, the prediction and determination of load reserve capacity is also important. In order to adapt to new forms of power systems, a day-ahead automatic generation control (AGC) reserve capacity demand prediction method based on the Fourier transform and the attention mechanism combined with a bidirectional long and short-term memory neural network model (Attention-BiLSTM) optimized by an improved whale optimization algorithm (IWOA) is proposed. Firstly, based on the response time, Fourier transform is used to refine the distinction between various types of load reserve demand, and the power of the AGC reserve band is calculated using Parseval's theorem to obtain the reserve capacity demand sequence. The maximum mutual information coefficient method is used to explore the relevant influencing factors of the AGC reserve sequence concerning the data characteristics of the AGC reserve sequence. Then, the historical daily AGC reserve demand sequences with relevant features are input into the Attention-BiLSTM prediction model, and the improved whale algorithm is used to automatically find the optimal hyperparameters to obtain better prediction results. Finally, the arithmetic simulation results show that the model proposed in this paper has the best prediction performance with the upper (0.8810) and lower (0.6651) bounds of the coefficient of determination ( R 2 ) higher than the other models, and it has the smallest mean absolute percentage error ( M A P E ) and root mean square error ( R M S E ). [ABSTRACT FROM AUTHOR]

In this paper, estimates are obtained for the Fourier transform of smooth charges (measures) concentrated on some nonconvex hypersurfaces. The summability of the maximal Randall function is proved for a wide class of nonconvex hypersurfaces. In addition, in the three-dimensional case, estimates are obtained depending on the Varchenko height. The accuracy of the obtained estimates is proved. The proof of the estimate for oscillatory integrals is based on the Weierstrass preparatory theorem. [ABSTRACT FROM AUTHOR]